Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}2x+9y &= 1 \\ -6x-9y &= 5\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Add the top and bottom equations. $-4x = 6$ Divide both sides by $-4$ and reduce as necessary. $x = -\dfrac{3}{2}$ Substitute $-\dfrac{3}{2}$ for $x$ in the top equation. $2( -\dfrac{3}{2})+9y = 1$ $-3+9y = 1$ $9y = 4$ $y = \dfrac{4}{9}$ The solution is $\enspace x = -\dfrac{3}{2}, \enspace y = \dfrac{4}{9}$.